How does the Turbulence Force Field work?
A force field is a vector field, meaning at every point in space it defines a vector based on some vector-valued function.  In the case of a force field, the vectors represent forces pushing the particles around.   In other words, at each frame the force field exerts a force on every particle dependant on the particle's position.

What differs between the different force fields is the vector function they are based on.  For a turbulence field it uses some noise pattern (Perlin, I think), creating random "turbulent" particle movement.  Most of the settings for the force field (the ones that change when you change the force field type) modify this noise pattern.

TL;DR: The turbulence force field pushes particles around based on their position in a 3D noise.



Here's a rough explanation of the key settings:


Strength controls the amount of force the field exerts on the particles.  Note that this is a multiplier to the strength given by the noise texture so it' actually a maximum value since the noise gives values from 0 to 1.
Size is the scale of the noise texture.  Larger values (compared to the scale of your scene) will make the particles move in bigger masses, while small values will produce more chaotic movement.
Noise is the amount distortion applied to the noise texture.  Higher values will make the particle movement tighter and more erratic.
Falloff - Type (Sphere/Tube/Cone) changes the shape of the falloff zone.
Falloff - Power changes the polynomial power, essentially the steepness, of the falloff.  A value of 0 here means no falloff, meaning all particles are affected by the force field equally no matter their relation to it.
Falloff - Minimum/Maximum changes the start and end point of the falloff curve.  The force field is in full effect for all particles closer than the minimum and does not effect particles beyond the maximum at all.